In practice, almost all signals are contaminated to certain degree by interfering signals and background noise. Since the test environment is usually non-ideal, contamination in data collection is unavoidable. Meanwhile, the test environment is typically arbitrary, and interfering signals and background noise are unknown a priori. As such, it is extremely challenging to extract target information from a mixture of data that include some unknown number of interfering signals and random background noise.
Common signal processing technologies such as the Fourier transform and short-time Fourier transform do not work, because they cannot handle the signals that are varying in time and frequency in an arbitrary manner. The wavelet transform allows for converting an arbitrary time signal into a form that either makes certain features of the original signal more amenable to study or enables the original data set to be described more succinctly. However, the standard discrete wavelet transform cannot be used directly to extract desired information from the measured data. This is because the discrete wavelet transform decomposes an arbitrary time signal into various scales, and displays their corresponding behaviors in the time domain at these scales. Such a treatment makes it difficult to analyze the behaviors of various components and features of the measured data, which is critical for extracting the desired information in practice.